id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-d960facd57e40c2a3a08e8ad82ef4d362b5f966c094d381dac2099993a3991b9 | A matrix has $N$ rows and $2k-1$ columns. Each column is filled with $M$ ones and $N-M$ zeros. A given row $i$ is "cool" if $\sum_{j=1}^{2k-1}a_{ij} \geq k$. Find the minimum and maximum number of cool rows for given $N$, $k$, and $M$. | [
"Minimum = 1, Maximum = 3"
] | [
"A matrix has $N$ rows and $2k-1$ columns. Each column is filled with $M$ ones and $N-M$ zeros. A given row $i$ is \"cool\" if $\\sum_{j=1}^{2k-1}a_{ij} \\geq k$. Find the minimum and maximum number of cool rows for given $N$, $k$, and $M$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fcda5a3555f49d21ade3684dd350e6861c62aaeb8ce6bad900c52f720d2f8f6d | A matrix with complex coefficients was squared. It turns out that this matrix and the matrix transposed to the original matrix are mutually inverse. Find $\mathrm{Re}(\mathrm{det}A)$ if this number is negative. | [
"\\(\\frac{-1}{2}\\)"
] | [
"A matrix with complex coefficients was squared. It turns out that this matrix and the matrix transposed to the original matrix are mutually inverse. Find $\\mathrm{Re}(\\mathrm{det}A)$ if this number is negative."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 7
} | 27 | 0 | 0 |
omr-ws-13a8b54ac09ed537885f97d394a00d168461f1ebb3d07654c1c392105afe2980 | A meadow is shaped as a square $12 \times 12$ km and is divided into three strips, each $4$ km wide. One strip is covered with snow, another with sand, and the middle strip is a skating rink. The skater runs at speeds of $12$ km/h on ice, $4$ km/h on snow, and $3$ km/h on sand. The skier runs at speeds of $3$ km/h on i... | [
"\\text{the athlete}"
] | [
"A meadow is shaped as a square $12 \\times 12$ km and is divided into three strips, each $4$ km wide. One strip is covered with snow, another with sand, and the middle strip is a skating rink. The skater runs at speeds of $12$ km/h on ice, $4$ km/h on snow, and $3$ km/h on sand. The skier runs at speeds of $3$ km/... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-ca10a3fe03863a644ebcf0a562d7084a943e2ff3ca93f1eb4c34ccdfdb35072f | A median of triangle \(ABC\) divides it into two triangles of equal perimeter. The length of this median is 21 and the length of the side to which it is drawn is 56. What is the perimeter of \(\triangle ABC\)? | [
"126"
] | [
"A median of triangle \\(ABC\\) divides it into two triangles of equal perimeter. The length of this median is 21 and the length of the side to which it is drawn is 56. What is the perimeter of \\(\\triangle ABC\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-724f2af36175904f091d8769f2d9c03af01135caf28e568e08c4878aa040f624 | A medical clinic wishes to have each provider see 32 patients per day. The clinic experiences a walk-in rate and a missed appointment rate that varies between 25% and 35%. Given the average data: - Scheduled: 33.1 patients - Walk-in: 5.6 patients - Seen: 27.1 patients - Missed: -11.6 patients Determine a formula to cal... | [
"S = \\frac{32}{1 - m + w}"
] | [
"A medical clinic wishes to have each provider see 32 patients per day. The clinic experiences a walk-in rate and a missed appointment rate that varies between 25% and 35%. Given the average data:\n- Scheduled: 33.1 patients\n- Walk-in: 5.6 patients\n- Seen: 27.1 patients\n- Missed: -11.6 patients\n\nDetermine a fo... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-441f4aa785ba6e275f4d879bf6b72dfb3308dba5d3014be13fdea41d64346e2d | A medical student serves a hospital 5 days a month, but he cannot serve on two consecutive days. In how many ways can he serve the hospital in January? | [
"$\\binom{27}{5}$"
] | [
"A medical student serves a hospital 5 days a month, but he cannot serve on two consecutive days. In how many ways can he serve the hospital in January?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8df68aa1fb7b981553eb927973d0d017f56751e06b7e5cde347474b367af9680 | A meeting has $n$ people ($n > 5$). We know that there exists a positive integer $m$ such that in any arbitrary $n-2$ people, there are exactly $3^m$ pairs of people that know each other. Find all possible values of $n$. | [
"7"
] | [
"A meeting has $n$ people ($n > 5$). We know that there exists a positive integer $m$ such that in any arbitrary $n-2$ people, there are exactly $3^m$ pairs of people that know each other. Find all possible values of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-521a6356afc6a8b2fdc66f1161ade6b73d494cb5c3c46a9e4e7300eeacc13644 | A meeting is held at a round table. It is known that 7 women have a woman on their right side, and 12 women have a man on their right side. It is also known that 75% of the men have a woman on their right side. How many people are sitting at the round table? | [
"35"
] | [
"A meeting is held at a round table. It is known that 7 women have a woman on their right side, and 12 women have a man on their right side. It is also known that 75% of the men have a woman on their right side. How many people are sitting at the round table?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-7c8cb81c59fa64d0bf13d2600ed5dd8e9834dba827883a87c56a641e14d71c4f | A meeting of 40 people must choose a chairman, a secretary and 5 members of a committee. How many different committees can be formed? | [
"$40\\cdot 39\\cdot C_{38}^5$"
] | [
"A meeting of 40 people must choose a chairman, a secretary and 5 members of a committee. How many different committees can be formed?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e567d29766d63717e0a902e8a799b73ec7195d7a04756a9d48aa3ca72e6f9594 | A meeting was attended by $n$ people who occupy $k$ tables, where $k \le \frac{n}{2}$. Each table is occupied by at least two people. The moderator selects two people from each table as representatives. Let $A$ be the number of ways to choose these representatives. Determine the maximum value of $A$ that is possible. | [
"\\dbinom{\\left\\lfloor \\frac{n}{k} \\right\\rfloor + 1}{2}^{n \\bmod k} \\cdot \\dbinom{\\left\\lfloor \\frac{n}{k} \\right\\rfloor}{2}^{k - (n \\bmod k)}"
] | [
"A meeting was attended by $n$ people who occupy $k$ tables, where $k \\le \\frac{n}{2}$. Each table is occupied by at least two people. The moderator selects two people from each table as representatives. Let $A$ be the number of ways to choose these representatives. Determine the maximum value of $A$ that is poss... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-db1fc2f73cd8270faad078fe8ac483cfbbfdc02be4dcc6e7b584f61e08f9196e | A merchant buys goods at $25\%$ off the list price. He desires to mark the goods so that he can give a discount of $20\%$ on the marked price and still clear a profit of $25\%$ on the selling price. What percent of the list price must he mark the goods? | [
"125\\%"
] | [
"A merchant buys goods at $25\\%$ off the list price. He desires to mark the goods so that he can give a discount of $20\\%$ on the marked price and still clear a profit of $25\\%$ on the selling price. What percent of the list price must he mark the goods?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-62f105016358c87f5716c95a3aa29514875ba37e00765f290756571535708918 | A merchant buys goods at 25% of the list price. He desires to mark the goods so that he can give a discount of 20% on the marked price and still clear a profit of 25% on the selling price. What percent of the list price must he mark the goods? | [
"75%"
] | [
"A merchant buys goods at 25% of the list price. He desires to mark the goods so that he can give a discount of 20% on the marked price and still clear a profit of 25% on the selling price. What percent of the list price must he mark the goods?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-94cec6cf1fd70bb5c9fe70634ced6d9dd3e7b649dac799e77bc1c556636c2045 | A merchant buys goods at 25% off the list price. He desires to mark the goods so that he can give a discount of 20% on the marked price and still make a profit of 25% on the selling price. What per cent of the list price must he mark the goods? | [
"117%"
] | [
"A merchant buys goods at 25% off the list price. He desires to mark the goods so that he can give a discount of 20% on the marked price and still make a profit of 25% on the selling price. What per cent of the list price must he mark the goods?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-72e01b40a0f94bb7c9301ca79be4ab39a940d542bb5a2c91c05e30bdd961ca76 | A merchant buys some oranges at the price of 3 for 40 cents and twice as many at the price of 5 for 84 cents. At what price per dozen must he sell the entire lot in order to break even? | [
"188 cents"
] | [
"A merchant buys some oranges at the price of 3 for 40 cents and twice as many at the price of 5 for 84 cents. At what price per dozen must he sell the entire lot in order to break even?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 20
} | 42 | 0 | 0 |
omr-ws-467ef8315bcc7bd7d6c9efcee264efb7be683e8f1ed0bd3eb3a0bb6acfde229f | A merchant has 6 bottles with volumes 16 oz, 18 oz, 22 oz, 24 oz, 32 oz, and 34 oz. Some are filled with orange juice, some with cherry juice, and one is empty. There is twice as much orange juice as cherry juice. What is the volume of the empty bottle? | [
"32 oz"
] | [
"A merchant has 6 bottles with volumes 16 oz, 18 oz, 22 oz, 24 oz, 32 oz, and 34 oz. Some are filled with orange juice, some with cherry juice, and one is empty. There is twice as much orange juice as cherry juice. What is the volume of the empty bottle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-501f7e0c74bc6ec67b38c8a8b6da4138c81dbf7b023b3640aa0ffc1beade4a1b | A merchant has coffee worth $20 a pound that they wish to mix with 40 lbs of coffee worth $80 a pound to get a mixture that can be sold for $60 a pound. How many pounds of the $20 coffee should be used? | [
"20"
] | [
"A merchant has coffee worth $20 a pound that they wish to mix with 40 lbs of coffee worth $80 a pound to get a mixture that can be sold for $60 a pound. How many pounds of the $20 coffee should be used?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e864039f794ea0b8f4badead262f2de6d91b2a720baf51515ed42925083da200 | A merchant is carrying money from point A to point B. There are robbers on the roads who rob travelers: on one road the robbers take $10\%$ of the amount currently available, on the other - $20\%$, etc. How should the merchant travel to bring as much of the money as possible to B? What part of the original amount will ... | [
"\\dfrac{18}{25}"
] | [
"A merchant is carrying money from point A to point B. There are robbers on the roads who rob travelers: on one road the robbers take $10\\%$ of the amount currently available, on the other - $20\\%$, etc. How should the merchant travel to bring as much of the money as possible to B? What part of the original amoun... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-b56dcd42bfc673153a4a1218fe93c38cea1aece1b906a4d501f8037e5c9bef24 | A merchant offers a large group of items at $30\%$ off. Later, the merchant takes $20\%$ off these sale prices and claims that the final price of these items is $50\%$ off the original price. As a percentage of the original price, what is the difference between the true discount and the merchant's claimed discount? (Yo... | [
"6\\%"
] | [
"A merchant offers a large group of items at $30\\%$ off. Later, the merchant takes $20\\%$ off these sale prices and claims that the final price of these items is $50\\%$ off the original price. As a percentage of the original price, what is the difference between the true discount and the merchant's claimed dis... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3460726d05bb9ab24916fbcd0e71d2789008d613375294cdcd721d65621e19e6 | A merganser mates every 7th day, a scaup mates every 11th day, and a gadwall mates every 13th day. A merganser, scaup, and gadwall all mate on Day 0. On Days N, N+1, and N+2, the merganser, scaup, and gadwall mate in some order with no two birds mating on the same day. Determine the smallest possible value of N. | [
"208"
] | [
"A merganser mates every 7th day, a scaup mates every 11th day, and a gadwall mates every 13th day. A merganser, scaup, and gadwall all mate on Day 0. On Days N, N+1, and N+2, the merganser, scaup, and gadwall mate in some order with no two birds mating on the same day. Determine the smallest possible value of N."
... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6b32bcd73a95884521a6d4253ac591f879eb5cc90b536320bf3f1f478cb27324 | A message consisting of \( n \) binary numbers is sent twice. If the two sequences coincide, the message is assumed to be correct. Find the probability that the message received is actually distorted, given that both sequences coincide. | [
"\\(\\frac{p^n}{q^n + p^n}\\)"
] | [
"A message consisting of \\( n \\) binary numbers is sent twice. If the two sequences coincide, the message is assumed to be correct. Find the probability that the message received is actually distorted, given that both sequences coincide."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-65740c18dd4fbb6105263d83408a9f6168f0f8899cff9812a3a72b46c932d9c6 | A messenger sets out on a moped from town $A$ to town $B$ located 120 km from $A$. An hour later, a second messenger sets out on a motor-cycle from $A$. He overtakes the first messenger, passes a commission to him and immediately starts back at the same speed. He arrives at $A$ at the moment the first messenger arrives... | [
"30 km/h"
] | [
"A messenger sets out on a moped from town $A$ to town $B$ located 120 km from $A$. An hour later, a second messenger sets out on a motor-cycle from $A$. He overtakes the first messenger, passes a commission to him and immediately starts back at the same speed. He arrives at $A$ at the moment the first messenger ar... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4264244eeedd4111aefc54c019811d5e3b3e1690cdb07660b6f39c46e2c67c95 | A metal cone of height 12 cm and radius 4 cm is lowered into a cylinder of radius 4 cm and height 100 cm, which is filled with water to a depth of 50 cm. The cone is lowered at a constant rate of 4 cm per second. At what rate is the water in the cylinder rising when the vertex of the cone is immersed to a depth of 6 cm... | [
"1 cm/s"
] | [
"A metal cone of height 12 cm and radius 4 cm is lowered into a cylinder of radius 4 cm and height 100 cm, which is filled with water to a depth of 50 cm. The cone is lowered at a constant rate of 4 cm per second. At what rate is the water in the cylinder rising when the vertex of the cone is immersed to a depth of... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 3,
"tir": 20
} | 33 | 0 | 0 |
omr-ws-1cc7065b1701ee97b147934e56f4d09a581e4ff0418a6e0abaf649b58063d573 | A metal cube with side length 1 meter has a hole drilled through its center, starting at one corner and ending at the opposite corner. If the radius of the drill is significantly less than 1 meter, what is the volume of the remaining material? | [
"1 - \\sqrt{3} \\pi r^2"
] | [
"A metal cube with side length 1 meter has a hole drilled through its center, starting at one corner and ending at the opposite corner. If the radius of the drill is significantly less than 1 meter, what is the volume of the remaining material?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 2
} | 15 | 0 | 0 |
omr-ws-ce651d499ae6f23b2beb4178d0d4d0fa7cea6d5904d936db7084d3f9ec7c9f72 | A metal door is made up of two parts: an Iron part and a Zinc part. Each cm of the Iron part weighs 9 kg, and each cm of the Zinc part weighs 15 kg. The Zinc part is 11 cm shorter than the Iron part. The door weighs 840 kg in total. How long is the Iron part? Given equation to fill in the blanks: \[ 9(x - 11) + ( \quad... | [
"41.875"
] | [
"A metal door is made up of two parts: an Iron part and a Zinc part. Each cm of the Iron part weighs 9 kg, and each cm of the Zinc part weighs 15 kg. The Zinc part is 11 cm shorter than the Iron part. The door weighs 840 kg in total. How long is the Iron part?\n\nGiven equation to fill in the blanks:\n\\[ 9(x - 11)... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-fd9470decf32d779dd505c71ac9eb858acbc9428710d0b48d132b5fe8ec2efbf | A metal flagpole on level ground broke at a point with the top part toppling over and the tip hitting the ground at a point 20 ft. from the base of the flagpole. It was rewelded but broke again at a point 5 ft. lower than the previous break point, with the top hitting the ground at a point 30 ft. from the base of the f... | [
"50"
] | [
"A metal flagpole on level ground broke at a point with the top part toppling over and the tip hitting the ground at a point 20 ft. from the base of the flagpole. It was rewelded but broke again at a point 5 ft. lower than the previous break point, with the top hitting the ground at a point 30 ft. from the base of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 19,
"tir": 3
} | 22 | 0 | 0 |
omr-ws-85968815b686fb464315299b8844bfd5c54c4c762e6640c2bed2b7bf9021ea8e | A metal rod has been broken into 3 pieces. The two break points are chosen randomly. What is the probability that an acute-angled triangle can be formed from the 3 pieces? | [
"\\ln 2 - \\frac{1}{2}"
] | [
"A metal rod has been broken into 3 pieces. The two break points are chosen randomly. What is the probability that an acute-angled triangle can be formed from the 3 pieces?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-af2aed211440b6d65f8348d0fc26c166f35cee7d264e084f8683c55118529157 | A meteor in the shape of a regular tetrahedron has an initial side length of 1. At time \( t = 1 \) second, its surface area is half of the original. If its side length is directly proportional to elapsed time, find \( k \) such that at time \( t = k \), its volume will be half of the original. | [
"\\( k = \\left( \\frac{1}{2} \\right)^{1/6} \\)"
] | [
"A meteor in the shape of a regular tetrahedron has an initial side length of 1. At time \\( t = 1 \\) second, its surface area is half of the original. If its side length is directly proportional to elapsed time, find \\( k \\) such that at time \\( t = k \\), its volume will be half of the original."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 4,
"tir": 10
} | 24 | 0 | 0 |
omr-ws-65d0239e617e8c20307a71e7f3cdc747ed6f9516b3df30d1fe1458a1dc2bce7b | A meteorology researcher is recording weather records in \( n \) cities for 10 days. For each day, a city's weather can be "Sunny" or "Rainy." We are told that for any two different days, there are equally many cities that have the same weather on those two days as the ones that have different weathers on those two day... | [
"\\( n \\) must be a multiple of 1024."
] | [
"A meteorology researcher is recording weather records in \\( n \\) cities for 10 days. For each day, a city's weather can be \"Sunny\" or \"Rainy.\" We are told that for any two different days, there are equally many cities that have the same weather on those two days as the ones that have different weathers on th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-966e043f4bdefc5bc282624e25dbe90b9cb345788c41376cfaa40ba0e4cdcf8c | A meter stick is cut in two random places, producing 3 segments. What is the probability that the longest segment is more than 50 centimeters long? | [
"\\(\\frac{3}{4}\\)"
] | [
"A meter stick is cut in two random places, producing 3 segments. What is the probability that the longest segment is more than 50 centimeters long?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-41522a524d4f0323d6cac2edd33e702b5a0031c8f60441857660e680c91234de | A meter stick is cut in two random places, producing 3 segments. What is the probability that the longest segment is more than 50 cm long? | [
"\\dfrac{3}{4}"
] | [
"A meter stick is cut in two random places, producing 3 segments. What is the probability that the longest segment is more than 50 cm long?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-75181de2a3d60d1e2a0eb19d46ad8336c2623e2105f47d8a6cc588fbc319d5da | A microbe either divides into two identical copies of itself or dies. If the probability of division is \( p = \frac{1}{3} \), what is the probability that the descendants of the microbe (the microbe colony) will eventually become extinct? | [
"1"
] | [
"A microbe either divides into two identical copies of itself or dies. If the probability of division is \\( p = \\frac{1}{3} \\), what is the probability that the descendants of the microbe (the microbe colony) will eventually become extinct?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9a294ad3706943c2d1d96a03674bf8ada3e960ad2bee8ecf5f412c9b02aa5cd9 | A mill produces three times as many units of plywood as lumber. The profit per unit of lumber is $20, and the profit per unit of plywood is $45. How many units of each must be produced and sold to make a profit of $15965? Provide the equations needed to solve this problem. | [
"\\( L = 103 \\), \\( P = 309 \\)"
] | [
"A mill produces three times as many units of plywood as lumber. The profit per unit of lumber is $20, and the profit per unit of plywood is $45. How many units of each must be produced and sold to make a profit of $15965? Provide the equations needed to solve this problem."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 18
} | 28 | 0 | 0 |
omr-ws-57eeec2dc12894026c8f04afe306a3b738d6b8ab6b84d017eb0dc29c8a026b76 | A mine-field is laid so that the mines are placed every 100 m along a straight line. What is the probability that a ship 20 m wide will blow up if it passes through the mine field at right angles to the line? | [
"0.2"
] | [
"A mine-field is laid so that the mines are placed every 100 m along a straight line. What is the probability that a ship 20 m wide will blow up if it passes through the mine field at right angles to the line?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d8b40c2a31b535cd1acdde96b0780343a415684c2fbf050cb1bc0e2cd227946a | A miniature piano takes 28 cubic centimeters of wood and 1.6 milliliters of paint. If the piano is scaled up by a factor of 1.5, how much wood and paint will the larger model need? | [
"94.5 cubic centimeters of wood and 3.6 milliliters of paint."
] | [
"A miniature piano takes 28 cubic centimeters of wood and 1.6 milliliters of paint. If the piano is scaled up by a factor of 1.5, how much wood and paint will the larger model need?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 19
} | 22 | 0 | 0 |
omr-ws-1f65125fc9f0ebd7d84a9e71371ae94b01d240873b415006729a87b5b28a9aa1 | A miraculous triple is a triple of positive integers $(a, b, c)$ satisfying: \[ \sum_{k=1}^{2014} \frac{\sin x}{\sin{kx} \sin{(kx + x)}} = \frac{\sin ax}{\sin{bx} \sin{cx}} \] for all real values of $x$ with $0 < x < \frac{\pi}{2015}$. The authors of this contest are certain they know the miraculous triple $(a_0, b_0, ... | [
"12176649"
] | [
"A miraculous triple is a triple of positive integers $(a, b, c)$ satisfying:\n\n\\[\n\\sum_{k=1}^{2014} \\frac{\\sin x}{\\sin{kx} \\sin{(kx + x)}} = \\frac{\\sin ax}{\\sin{bx} \\sin{cx}}\n\\]\n\nfor all real values of $x$ with $0 < x < \\frac{\\pi}{2015}$. The authors of this contest are certain they know the mira... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-ef5618c8522d46c9871a128fd24148e9584b84735e4d34dc2d3e2cb7267bb4a7 | A mirror has an area of 1 m² and is shaped like a rectangle with a semicircle at each end. The frame around the mirror costs twice as much per square centimeter for the curved part compared to the straight part. Determine the radius of the semicircles that minimizes the cost of the frame. | [
"\\( \\frac{1}{\\sqrt{3\\pi}} \\)"
] | [
"A mirror has an area of 1 m² and is shaped like a rectangle with a semicircle at each end. The frame around the mirror costs twice as much per square centimeter for the curved part compared to the straight part. Determine the radius of the semicircles that minimizes the cost of the frame."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-5a7bef4e7784f81564b5fe1cf3b91db4577e17a752e37fae86b01215a9cb561b | A mirror is constructed in the shape of $y=\pm{\sqrt{x}}$ for $0\le{x}\le{1}$ and $y=\pm{1}$ for $1<x<9$. A ray of light enters at $(10,1)$ with slope $1$. How many times does it bounce before leaving? | [
"12"
] | [
"A mirror is constructed in the shape of $y=\\pm{\\sqrt{x}}$ for $0\\le{x}\\le{1}$ and $y=\\pm{1}$ for $1<x<9$. A ray of light enters at $(10,1)$ with slope $1$. How many times does it bounce before leaving?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f8f60fc9653a51c7d04b02d039d482518a98f5704c466b42ef4849e66930a449 | A mirror is in the shape of the curve \( x = \frac{1}{4} y^2 \). A light shines at the mirror along the line \( y = h \) from some large \( x \)-value. The reflected beam of light passes through the point \( (1,0) \). Calculate the \( y \)-coordinate of the point on the mirror where the light hits, in terms of \( h \). | [
"h"
] | [
"A mirror is in the shape of the curve \\( x = \\frac{1}{4} y^2 \\). A light shines at the mirror along the line \\( y = h \\) from some large \\( x \\)-value. The reflected beam of light passes through the point \\( (1,0) \\). Calculate the \\( y \\)-coordinate of the point on the mirror where the light hits, in t... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dbd38d9cced27cb0a310d33d99f74a295d8f4c8fef793c82f5c75f66868edb6d | A missile $\mathcal{E}$ is designed to follow a target with a constant speed such that the vector of the missile's velocity is always directed toward the target. The target is an $F-14$ plane moving in a straight line with speed $V'$. The speed of the missile $\mathcal{E}$ is $V$, where $V' \ge V$. Determine the points... | [
"\\text{No interception}"
] | [
"A missile $\\mathcal{E}$ is designed to follow a target with a constant speed such that the vector of the missile's velocity is always directed toward the target. The target is an $F-14$ plane moving in a straight line with speed $V'$. The speed of the missile $\\mathcal{E}$ is $V$, where $V' \\ge V$. Determine th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 1
} | 13 | 0 | 0 |
omr-ws-809e92370bd7bd1379fdf5b26598fc4c13c4ef16dac96efcfe0c9b636b9c80e5 | A missile is traveling along the curve $\vec{r}(t) = t\vec{i} + t^2\vec{j} + t^3\vec{k}$. Dylan wants to shoot a laser so that the laser will be tangent to the missile at the point (2,4,8). Where should Dylan position himself on the ground (xy-plane)? You may assume that the earth is flat. | [
"\\(\\left(\\frac{4}{3}, \\frac{4}{3}\\right)\\)"
] | [
"A missile is traveling along the curve $\\vec{r}(t) = t\\vec{i} + t^2\\vec{j} + t^3\\vec{k}$. Dylan wants to shoot a laser so that the laser will be tangent to the missile at the point (2,4,8). Where should Dylan position himself on the ground (xy-plane)? You may assume that the earth is flat."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-e3d5778747bd8e3257054f9fa602f01f08371b2013ed153b062750f526265409 | A mobile point $P$ proceeds from the origin $O$ in the $xy$ plane according to the following rule: when a dice is rolled $n$ times, if the spot is a multiple of 3, $P$ proceeds one unit in the positive $x$ direction, and if the spot is not a multiple of 3, $P$ proceeds two units in the positive $y$ direction. Let $P_n$... | [
"\\(\\frac{17n^2 + 10n}{9}\\)"
] | [
"A mobile point $P$ proceeds from the origin $O$ in the $xy$ plane according to the following rule: when a dice is rolled $n$ times, if the spot is a multiple of 3, $P$ proceeds one unit in the positive $x$ direction, and if the spot is not a multiple of 3, $P$ proceeds two units in the positive $y$ direction. Let ... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-168bf5b0a002d391faa219813476d29d8fcaa7affd85da7521d16bd906f83f7e | A model car is tested on a closed circuit 600 meters long, consisting of flat stretches, uphill, and downhill sections. The car's velocity depends only on the type of stretch it is driving on, with velocities $v_s$, $v_p$, and $v_d$ for uphill, flat, and downhill sections, respectively, where $v_s < v_p < v_d$. All vel... | [
"(10, 12, 15)"
] | [
"A model car is tested on a closed circuit 600 meters long, consisting of flat stretches, uphill, and downhill sections. The car's velocity depends only on the type of stretch it is driving on, with velocities $v_s$, $v_p$, and $v_d$ for uphill, flat, and downhill sections, respectively, where $v_s < v_p < v_d$. Al... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b092b09080b50b33b4553a2eda4e0e18e0ccb8a2a3a3a4c8f984040bec12f59e | A model car is tested on a closed circuit that is 600 meters long, consisting of flat stretches, uphill, and downhill sections. The velocities of the car on these sections are $v_s$, $v_p$, and $v_d$ respectively, where $v_s < v_p < v_d$ and all velocities are integers in meters per second. The time taken to complete t... | [
"\\( (v_s, v_p, v_d) = (10, 12, 15), (9, 12, 18), (8, 12, 24), (7, 12, 42) \\)"
] | [
"A model car is tested on a closed circuit that is 600 meters long, consisting of flat stretches, uphill, and downhill sections. The velocities of the car on these sections are $v_s$, $v_p$, and $v_d$ respectively, where $v_s < v_p < v_d$ and all velocities are integers in meters per second. The time taken to compl... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7d52ddecb6e28cf82e8e135d2d605485dbb5d5ccd458bb4a05d63956afd1f295 | A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is \(a(t) = 60t\), at which time the fuel is exhausted and it becomes a freely “falling” body. Fourteen seconds later, the rocket’s parachute opens, and the (downward) velocity slows linearly to \(-18 \frac{\text{ft}}{\tex... | [
"\\frac{410}{9} \\text{ seconds}"
] | [
"A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is \\(a(t) = 60t\\), at which time the fuel is exhausted and it becomes a freely “falling” body. Fourteen seconds later, the rocket’s parachute opens, and the (downward) velocity slows linearly to \\(-18 \\frac{\\text... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-90daa9bbb2deee05f87c501a782e7a136bff3751cd4ddb637930fc53675ad4c6 | A modified knight moves $p$ fields horizontally or vertically and $q$ fields in the perpendicular direction on an infinite chessboard. If the knight returns to its initial field after $n$ moves, calculate the smallest possible even value of $n$ for $p=2$ and $q=3$. | [
"2"
] | [
"A modified knight moves $p$ fields horizontally or vertically and $q$ fields in the perpendicular direction on an infinite chessboard. If the knight returns to its initial field after $n$ moves, calculate the smallest possible even value of $n$ for $p=2$ and $q=3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-778a000f78b5ceb3c7ee2878666a956adb88943cc5e64fea562e373ac2227a3c | A mom of 3 kids told a math professor that the product of their ages is 72. The sum of their ages was also given, but the specific sum is unknown. The professor initially stated that this information was insufficient. However, after the mom mentioned that the oldest child has an iguana, the professor was able to determ... | [
"3, 3, 8"
] | [
"A mom of 3 kids told a math professor that the product of their ages is 72. The sum of their ages was also given, but the specific sum is unknown. The professor initially stated that this information was insufficient. However, after the mom mentioned that the oldest child has an iguana, the professor was able to d... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-a3ec802ca54eaab2bc13bf136b4f3d5afc5b6decb8d53219c5a23fe021f5c0e6 | A monic cubic polynomial $f(x)$ has roots $a^2, b^2$, and $c^2$. A triangle whose side lengths are $a, b,$ and $c$ has area 1. Given that $f\left(\frac{a^2+b^2+c^2}{4}\right)-f(0)=60$, find $f(10)-f(0)$. | [
"120"
] | [
"A monic cubic polynomial $f(x)$ has roots $a^2, b^2$, and $c^2$. A triangle whose side lengths are $a, b,$ and $c$ has area 1. Given that $f\\left(\\frac{a^2+b^2+c^2}{4}\\right)-f(0)=60$, find $f(10)-f(0)$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-438336a08cc0d1dfb21299097421574bb2762e1b344d777dff2ed9e2738646ff | A monic cubic polynomial \( P(x) \) has roots \( \alpha, \beta, \gamma \). Given that \( P'(\alpha) = 0 \), \( P'(\beta) = 0 \), \( P'(y) = 25 \), \( \alpha + \beta + y = -1 \), and \( \alpha \beta y = 12 \), find the roots of \( P(x) \). | [
"\\(-2, -2, 3\\)"
] | [
"A monic cubic polynomial \\( P(x) \\) has roots \\( \\alpha, \\beta, \\gamma \\). Given that \\( P'(\\alpha) = 0 \\), \\( P'(\\beta) = 0 \\), \\( P'(y) = 25 \\), \\( \\alpha + \\beta + y = -1 \\), and \\( \\alpha \\beta y = 12 \\), find the roots of \\( P(x) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 14
} | 17 | 0 | 0 |
omr-ws-bfb4bb294a129ec13d29352d93b96006e10076056a861dbfa7a69710ac28abba | A monic cubic polynomial has positive real roots \(a\), \(b\), and \(c\) such that the coefficient of the \(x^2\) term is greater than \(-10\). Another cubic polynomial is created with roots \(a+1\), \(b+1\), and \(c+1\). Find the integer that bounds the magnitude of the sum of the coefficients of the second cubic poly... | [
"38"
] | [
"A monic cubic polynomial has positive real roots \\(a\\), \\(b\\), and \\(c\\) such that the coefficient of the \\(x^2\\) term is greater than \\(-10\\). Another cubic polynomial is created with roots \\(a+1\\), \\(b+1\\), and \\(c+1\\). Find the integer that bounds the magnitude of the sum of the coefficients of ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-818d6b76958ccc723994f9427ae8b6a312e5b11f68d45edb5b8bb3c9237e70c0 | A monic cubic polynomial with integral coefficients has roots 3 and 5, and another root. After multiplying this cubic by a factor of the form \(x - b\), the resulting quartic polynomial has 6 as a root. If the original cubic polynomial has an \(x\)-coefficient of 8, what is the \(x\)-coefficient of the quartic polynomi... | [
"288"
] | [
"A monic cubic polynomial with integral coefficients has roots 3 and 5, and another root. After multiplying this cubic by a factor of the form \\(x - b\\), the resulting quartic polynomial has 6 as a root. If the original cubic polynomial has an \\(x\\)-coefficient of 8, what is the \\(x\\)-coefficient of the quart... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1f51144646c9c5c73f08ed562e6c2015fe695bf7febd90de8ff97ae2da00e056 | A monic polynomial $P(x)$ of minimal degree has roots $73$, $-47$, and $13$. If the sum of the coefficients of $P(x)$ can be expressed as $a^2 - n$, where $n$ is a minimal natural number and $a$ is a natural number, find $\frac{a}{n}$. | [
"\\(\\frac{17}{12}\\)"
] | [
"A monic polynomial $P(x)$ of minimal degree has roots $73$, $-47$, and $13$. If the sum of the coefficients of $P(x)$ can be expressed as $a^2 - n$, where $n$ is a minimal natural number and $a$ is a natural number, find $\\frac{a}{n}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-3860d2abfa3f14ee7f4f1cb0050e7b72c705aab36a67990dc612cf28b07a7215 | A monic polynomial \( P(x) \) of degree \( n \) takes on the integer values \( 1 \le P(x) \le n \) for \( 1 \le x \le n \) in random order. What is the expected value of \( P(n+1) \)? | [
"\\(\\frac{n+1}{2} + n!\\)"
] | [
"A monic polynomial \\( P(x) \\) of degree \\( n \\) takes on the integer values \\( 1 \\le P(x) \\le n \\) for \\( 1 \\le x \\le n \\) in random order. What is the expected value of \\( P(n+1) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 7
} | 53 | 0 | 0 |
omr-ws-4fd90c63673ece7c8eb59517afd0576b4d2571a9c52dd8cbbe00f6972f7f6521 | A monic polynomial with real coefficients and complex roots $\chi_1, \cdots, \chi_m$ (counted with multiplicity) is given. It is known that there does not exist a monic polynomial $Q$ with real coefficients such that the product $PQ$ is a "tasty" polynomial (i.e., all coefficients are in the range $[-1,1]$). Determine ... | [
"\\((1, \\infty)\\)"
] | [
"A monic polynomial with real coefficients and complex roots $\\chi_1, \\cdots, \\chi_m$ (counted with multiplicity) is given. It is known that there does not exist a monic polynomial $Q$ with real coefficients such that the product $PQ$ is a \"tasty\" polynomial (i.e., all coefficients are in the range $[-1,1]$). ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 6
} | 9 | 0 | 0 |
omr-ws-addc9f7904f248355133c44acf1a2a3eb35cb0de138e15dcad9527584eef5bdf | A monic quadratic $p(x)$ has $p(5) = 12$ and $p(p(3)) = 5$. If the quadratic $p(x)$ has the property that $p(2)$ is minimized, and $p(1)$ can be expressed as $a - \sqrt{b}$, find $a + b$. | [
"60"
] | [
"A monic quadratic $p(x)$ has $p(5) = 12$ and $p(p(3)) = 5$. If the quadratic $p(x)$ has the property that $p(2)$ is minimized, and $p(1)$ can be expressed as $a - \\sqrt{b}$, find $a + b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-d5c08c1123ccda0cbaf1bb97c54d7f0ba1ed443ea3efd801c135fa0384b3378b | A monic quadratic polynomial \( Q(x) = x^2 + ax + b \) has two roots \( r_1 < r_2 \). It is known that the sum \( r_1^0r_2^0 + r_1^1r_2^0 + r_1^1r_2^1 + r_1^2r_2^1 + r_1^2r_2^2 + \dots \) converges and evaluates to 1 less than the sum \( r_2^0r_1^0 + r_2^1r_1^0 + r_2^1r_1^1 + r_2^2r_1^1 + r_2^2r_1^2 + \dots \). Let \( ... | [
"1"
] | [
"A monic quadratic polynomial \\( Q(x) = x^2 + ax + b \\) has two roots \\( r_1 < r_2 \\). It is known that the sum \\( r_1^0r_2^0 + r_1^1r_2^0 + r_1^1r_2^1 + r_1^2r_2^1 + r_1^2r_2^2 + \\dots \\) converges and evaluates to 1 less than the sum \\( r_2^0r_1^0 + r_2^1r_1^0 + r_2^1r_1^1 + r_2^2r_1^1 + r_2^2r_1^2 + \\do... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f576aa9a185dba9bf411a9385143ad7ba51daab17cf6771006b714ca6b47c221 | A monic quadratic polynomial \( Q(x) = x^2 + ax + b \) has two roots \( r_1 < r_2 \). It is known that the sum \[ r_1^0 r_2^0 + r_1^1 r_2^0 + r_1^1 r_2^1 + r_1^2 r_2^1 + r_1^2 r_2^2 + \ldots \] converges and evaluates to one less than the sum \[ r_2^0 r_1^0 + r_2^1 r_1^0 + r_2^1 r_1^1 + r_2^2 r_1^1 + r_2^2 r_1^2 + \ldo... | [
"1"
] | [
"A monic quadratic polynomial \\( Q(x) = x^2 + ax + b \\) has two roots \\( r_1 < r_2 \\). It is known that the sum \n\\[ r_1^0 r_2^0 + r_1^1 r_2^0 + r_1^1 r_2^1 + r_1^2 r_2^1 + r_1^2 r_2^2 + \\ldots \\]\nconverges and evaluates to one less than the sum \n\\[ r_2^0 r_1^0 + r_2^1 r_1^0 + r_2^1 r_1^1 + r_2^2 r_1^1 + ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-df9b40e540111cf282882fcd1083b586f5c71aabf9827e273f74a3addde50f68 | A monic quartic polynomial $f(x)$ with four distinct complex roots $r_1$, $r_2$, $r_3$, and $r_4$ satisfies the following conditions: - $r_1 + r_2 + r_3 + r_4 = -20$ - $r_1r_2 + r_1r_3 + r_1r_4 + r_2r_3 + r_2r_4 + r_3r_4 = 0$ - $r_1r_2r_3 + r_1r_2r_4 + r_1r_3r_4 + r_2r_3r_4 = -1$ - $r_1r_2r_3r_4 = 20$ - $|r_1| \le |r_2... | [
"21"
] | [
"A monic quartic polynomial $f(x)$ with four distinct complex roots $r_1$, $r_2$, $r_3$, and $r_4$ satisfies the following conditions:\n- $r_1 + r_2 + r_3 + r_4 = -20$\n- $r_1r_2 + r_1r_3 + r_1r_4 + r_2r_3 + r_2r_4 + r_3r_4 = 0$\n- $r_1r_2r_3 + r_1r_2r_4 + r_1r_3r_4 + r_2r_3r_4 = -1$\n- $r_1r_2r_3r_4 = 20$\n- $|r_1... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-51c3c7fc83bcbbf35b263d8afcf4a0d73bb80a75b0e6da1d17cff2e87826bbb6 | A monkey is randomly typing on a keyboard with 27 keys (26 letters and a space key). What is the expected number of letters the monkey will type before the word "mathlinks" appears? | [
"\\(27^9 - 8\\)"
] | [
"A monkey is randomly typing on a keyboard with 27 keys (26 letters and a space key). What is the expected number of letters the monkey will type before the word \"mathlinks\" appears?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 17
} | 22 | 0 | 0 |
omr-ws-79c15c8a2a34c2f28489e90d1ba20b629eeb8c3ed04ec0dd35581436b515faf9 | A monkey must travel a path of length $n-1$, from $a_1$ to $a_n$. On every turn, he may jump any distance forward but cannot go backwards. How many times must he travel the path so that he has jumped at least once from $a_i$ to $a_j$ for all $1 \leq i < j \leq n$? (After reaching $a_n$, one travel is completed and he s... | [
"\\(n-1\\)"
] | [
"A monkey must travel a path of length $n-1$, from $a_1$ to $a_n$. On every turn, he may jump any distance forward but cannot go backwards. How many times must he travel the path so that he has jumped at least once from $a_i$ to $a_j$ for all $1 \\leq i < j \\leq n$? (After reaching $a_n$, one travel is completed a... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-0912b06fa0ef7d912de6e1dce47f4be3bf30f1ffdd6a3158fa32a59353ca525f | A monkey sees two birds at angles x and y above and below the horizontal, respectively, with both birds a horizontal distance d away. How far apart are the birds vertically? | [
"d(tanx+tany)"
] | [
"A monkey sees two birds at angles x and y above and below the horizontal, respectively, with both birds a horizontal distance d away. How far apart are the birds vertically?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-8f8002e0642b417f6b428ce595575f4c4969604cda987b40eff71f021c2652d0 | A monkey tries to climb a greasy 100-foot high pole. Starting at the bottom, it climbs 1 foot, then slides down \( \frac{2}{3} \) of the distance it has just climbed, and then climbs \( \frac{5}{3} \) of the distance it has just slid down. The monkey thereafter repeats the process until, if ever, it reaches the top of ... | [
"31"
] | [
"A monkey tries to climb a greasy 100-foot high pole. Starting at the bottom, it climbs 1 foot, then slides down \\( \\frac{2}{3} \\) of the distance it has just climbed, and then climbs \\( \\frac{5}{3} \\) of the distance it has just slid down. The monkey thereafter repeats the process until, if ever, it reaches ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-15c1f7559c37473547db47964ead287e1313ba84c040533afe6257b4c79f6417 | A monkey tries to climb a slippery 10-foot high pole. Starting at the bottom, he climbs 5 feet and slides half way to the ground. He then climbs an additional 5 feet and again slides half-way to the ground. This process is repeated indefinitely unless the monkey reaches the top of the pole. (a) How high is the monkey a... | [
"\\( a_n = 5 \\cdot \\frac{2^n - 1}{2^{n-1}} \\)"
] | [
"A monkey tries to climb a slippery 10-foot high pole. Starting at the bottom, he climbs 5 feet and slides half way to the ground. He then climbs an additional 5 feet and again slides half-way to the ground. This process is repeated indefinitely unless the monkey reaches the top of the pole.\n(a) How high is the mo... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 12
} | 15 | 0 | 0 |
omr-ws-1c5fe2d5c846221e173d6568b007c9927b26c737af5f819d972bee4259b19d6d | A monkey types a text with n symbols on a keyboard containing z symbols, including the two parenthesis symbols ( and ). Each symbol is equally likely to be typed. Determine the probability that the text written by the monkey has the parentheses used correctly (properly). | [
"\\(\\frac{1}{z^n} \\sum_{k=0}^{\\lfloor n/2 \\rfloor} \\frac{(z-2)^{n-2k}}{k+1} \\binom{2k}{k}\\)"
] | [
"A monkey types a text with n symbols on a keyboard containing z symbols, including the two parenthesis symbols ( and ). Each symbol is equally likely to be typed. Determine the probability that the text written by the monkey has the parentheses used correctly (properly)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 44,
"genselect": 5,
"tir": 24
} | 73 | 0 | 0 |
omr-ws-165528c02ee56d63c881ac23f65d2190e3670dd54efa6f0bcbe62229c2bb39db | A monkey types on a 26-letter keyboard that has lowercase letters only. Each letter is chosen independently and uniformly at random from the alphabet. If the monkey types 1,000,000 letters, what is the expected number of times that the sequence "proof" appears? | [
"0.0842"
] | [
"A monkey types on a 26-letter keyboard that has lowercase letters only. Each letter is chosen independently and uniformly at random from the alphabet. If the monkey types 1,000,000 letters, what is the expected number of times that the sequence \"proof\" appears?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-888bae1d63d6d1dee7e25f236f2ce717455e29eaf73cf15fe713a6b3fce39fc4 | A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month? | [
"3"
] | [
"A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 31
} | 31 | 0 | 0 |
omr-ws-1e8cc3f7cc7423edf6308e3783f4fc17ee6343498d6440acc01fb6301a4a74dd | A monument consists of a flagpole 15 meters tall standing on a mound in the shape of a cone with a vertex angle of 140 degrees. How long is the shadow cast by the pole on the cone when the angle of elevation of the sun is 62 degrees? | [
"7.1"
] | [
"A monument consists of a flagpole 15 meters tall standing on a mound in the shape of a cone with a vertex angle of 140 degrees. How long is the shadow cast by the pole on the cone when the angle of elevation of the sun is 62 degrees?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-1008aa8c0538c31dfb6ec8b25a0170c2d2e07b3ca348e0c994629e6202cc384e | A moon revolves around a planet every 5 years, the planet revolves around a sun every 9 years, and the sun revolves around a greater star every 3 years. If they last aligned on September 15th, 2020, how many times will at least two of these celestial bodies align between September 15th, 2020, and September 15th, 2022? | [
"1"
] | [
"A moon revolves around a planet every 5 years, the planet revolves around a sun every 9 years, and the sun revolves around a greater star every 3 years. If they last aligned on September 15th, 2020, how many times will at least two of these celestial bodies align between September 15th, 2020, and September 15th, 2... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-5fec1d68a1a3a6508d3610d54058fe47b7549947d16270fcfc52f155e925d13e | A mortgage of \$100,000 is to be paid off in exactly 30 years. If the annual interest rate is 9.25%, compounded monthly, what will the monthly payment be? | [
"849.00"
] | [
"A mortgage of \\$100,000 is to be paid off in exactly 30 years. If the annual interest rate is 9.25%, compounded monthly, what will the monthly payment be?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aaceb54fc9110e55469eabd76d1120f85a22bb873ada7f3aefe465d21d86437d | A motel bills its customers by charging a flat fee for the first night and then adding on a fixed amount for every night thereafter. If it costs George $\$155$ to stay in the motel for 3 nights and Noah $\$290$ to stay in the motel for 6 nights, how much is the flat fee for the first night? | [
"\\$65"
] | [
"A motel bills its customers by charging a flat fee for the first night and then adding on a fixed amount for every night thereafter. If it costs George $\\$155$ to stay in the motel for 3 nights and Noah $\\$290$ to stay in the motel for 6 nights, how much is the flat fee for the first night?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2617f59e8c068ac82661bb9241fae046d5f7ce9301dc017037f2565bd882defb | A mothball in the shape of a sphere starts with a radius of 1 cm. The material evaporates at a rate proportional to the surface area. After one month, the radius is 0.5 cm. How many months from the start is it before the radius is 0.2 cm? | [
"4"
] | [
"A mothball in the shape of a sphere starts with a radius of 1 cm. The material evaporates at a rate proportional to the surface area. After one month, the radius is 0.5 cm. How many months from the start is it before the radius is 0.2 cm?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-58412ab4e3e430c10e0b62a4dc78ec08df4393126859fec6b2c0cf183cb0e304 | A mother and her son are playing a game with a $300g$ wheel of cheese and $280g$ of butter. The son divides the cheese into $4$ slices and the mother divides the butter between two plates. The son then places the cheese slices on the plates. The son wins if the amount of cheese on each plate is not less than the amount... | [
"The son can guarantee a win."
] | [
"A mother and her son are playing a game with a $300g$ wheel of cheese and $280g$ of butter. The son divides the cheese into $4$ slices and the mother divides the butter between two plates. The son then places the cheese slices on the plates. The son wins if the amount of cheese on each plate is not less than the a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 5,
"tir": 4
} | 12 | 0 | 0 |
omr-ws-f974507de59bdbe68f7daede694e15d11b4ba15c93d2435bb60b359f7a0a6bbe | A mother and her son are playing a game with a 300g wheel of cheese and 280g of butter. The son divides the cheese into 4 slices and the mother divides the butter between two plates. The son then places the cheese slices on the plates. The son wins if the amount of cheese on each plate is not less than the amount of bu... | [
"Yes"
] | [
"A mother and her son are playing a game with a 300g wheel of cheese and 280g of butter. The son divides the cheese into 4 slices and the mother divides the butter between two plates. The son then places the cheese slices on the plates. The son wins if the amount of cheese on each plate is not less than the amount ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-1ce2234bcbef8b0f4d8bf6178be20e7166b4bff09de5cde917c2a9720b602981 | A mother has three daughters whose ages are whole numbers and the product of their ages is 36. The sum of their ages is not enough information to determine their ages uniquely. However, it is known that the oldest daughter is named Annie. What are the ages of the daughters? | [
"2, 2, 9"
] | [
"A mother has three daughters whose ages are whole numbers and the product of their ages is 36. The sum of their ages is not enough information to determine their ages uniquely. However, it is known that the oldest daughter is named Annie. What are the ages of the daughters?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 32
} | 45 | 0 | 0 |
omr-ws-1b663c44581ef1f1ffec942653f8f0ee3379f873a114647ee79b91d4c5373c01 | A mother horse and a baby horse walk 10 miles from point A to point B. The mother’s speed is 10 miles per hour and the baby’s speed is 5 miles per hour. The mother rests for 2 minutes after every x minutes of walking, while the baby keeps walking without resting. They arrive together at the destination. Find the range ... | [
"2"
] | [
"A mother horse and a baby horse walk 10 miles from point A to point B. The mother’s speed is 10 miles per hour and the baby’s speed is 5 miles per hour. The mother rests for 2 minutes after every x minutes of walking, while the baby keeps walking without resting. They arrive together at the destination. Find the r... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-16bbfd9bb3ac42c418450e22b5398eab4b6ffc695fd082cb3e3dfba4e17d81b5 | A mother purchases 5 blue plates, 2 red plates, 2 green plates, and 1 orange plate. How many ways are there for her to arrange these plates for dinner around her circular table if she doesn't want the 2 green plates to be adjacent? | [
"588"
] | [
"A mother purchases 5 blue plates, 2 red plates, 2 green plates, and 1 orange plate. How many ways are there for her to arrange these plates for dinner around her circular table if she doesn't want the 2 green plates to be adjacent?\n"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-08d7cbe181f5fba65652d8487d2c2005106f5edccf858dec08a60884cdf12bdc | A mother travels down a 100 mile long river with her two children. It would take her 20 minutes to travel the entire length downstream, and 50 minutes to travel the entire length upstream. During the journey, one of her children was intrigued by a large rock in the middle of the river, and climbed out on it to be left ... | [
"41"
] | [
"A mother travels down a 100 mile long river with her two children. It would take her 20 minutes to travel the entire length downstream, and 50 minutes to travel the entire length upstream. During the journey, one of her children was intrigued by a large rock in the middle of the river, and climbed out on it to be ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7baf26eded1ebcef02bb7300222af66161d47ec2f29efba95d5fb019e3118854 | A mother wants to spend at least 40 dollars but no more than 60 dollars on school clothes for her child. Shirts cost 10 dollars and pants cost 20 dollars. How many of each could she buy? | [
"\\((4,0), (5,0), (6,0), (0,2), (0,3), (2,1), (3,1), (1,2), (4,1), (2,2)\\)"
] | [
"A mother wants to spend at least 40 dollars but no more than 60 dollars on school clothes for her child. Shirts cost 10 dollars and pants cost 20 dollars. How many of each could she buy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 23
} | 36 | 0 | 0 |
omr-ws-1a9d254530bff387a6f14d14932df7c28b84f79796327f97d46f29ee6e74fccf | A motor boat and a raft simultaneously start out downstream from $A$. The motor boat covers 96 km, turns back and arrives at $A$ in 14 hours. Find the speed of the motor boat in still water and the speed of the current if it is known that the two craft met at a distance of 24 km from $A$ when the motor boat was returni... | [
"u=14 \\text{ km/h}, v=2 \\text{ km/h}"
] | [
"A motor boat and a raft simultaneously start out downstream from $A$. The motor boat covers 96 km, turns back and arrives at $A$ in 14 hours. Find the speed of the motor boat in still water and the speed of the current if it is known that the two craft met at a distance of 24 km from $A$ when the motor boat was re... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-485f94bec56b45a16efb346d0e025337c8e475983de67c8df3a397787fd57e4d | A motor boat travels at 10 km/h relative to the water. The water flows at a rate of 10 km/h. What is the boat's velocity relative to the shore when it heads upstream? Downstream? | [
"Upstream: 0 km/h, Downstream: 20 km/h"
] | [
"A motor boat travels at 10 km/h relative to the water. The water flows at a rate of 10 km/h. What is the boat's velocity relative to the shore when it heads upstream? Downstream?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 27
} | 34 | 0 | 0 |
omr-ws-d3b1f8c1a7ecde94456c13c436884babbdb8e35c4d1af772909d19eac84aeeb9 | A motor boat went 14 kilometres downstream in 30 minutes. The return trip upstream took 6 minutes longer. Find the rate of the boat in still water and the rate of the current. | [
"\\(x = 18.2\\) km/h, \\(y = 9.8\\) km/h"
] | [
"A motor boat went 14 kilometres downstream in 30 minutes. The return trip upstream took 6 minutes longer. Find the rate of the boat in still water and the rate of the current."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fdfd05b26f717071336faa701f64d5a44e180c719a672642ef84ebfbcc7ff156 | A motor vehicle insurance company divides policyholders into three categories: high, medium, and low risk. The high-risk group contains 10% of people, each with a 35% chance of making a claim during a particular year. The medium-risk group contains 20% of people, each with a 15% chance of making a claim during a year. ... | [
"0.8075"
] | [
"A motor vehicle insurance company divides policyholders into three categories: high, medium, and low risk. The high-risk group contains 10% of people, each with a 35% chance of making a claim during a particular year. The medium-risk group contains 20% of people, each with a 15% chance of making a claim during a y... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f1c36ed844c0ea243fd1c1eb508556f0d4e29eceeb40b96a736ca52aa332c38c | A motor-boat left point $A$ to go against the stream of a river, and a raft started simultaneously from point $B$ situated upstream from point $A$. In a hours they met and continued moving without stops. Having reached $B$ the motor-boat, without any delay, turned, began its return trip, and caught up with the raft at ... | [
"$a\\left(1+\\sqrt 2\\right)$ hours"
] | [
"A motor-boat left point $A$ to go against the stream of a river, and a raft started simultaneously from point $B$ situated upstream from point $A$. In a hours they met and continued moving without stops. Having reached $B$ the motor-boat, without any delay, turned, began its return trip, and caught up with the raf... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e3a433ddf2a45e71fc4a8ae5638396cdc20398cf436931eafbf5a60ebbe157e8 | A motor-boat, whose speed in dead water is 6 km/h, starts from pier $A$ to pier $B$ which is 12 km down the river. At the same time a motor-launch whose speed in dead water is 10 km/h starts from $B$ to $A$. After their meeting, they turn back and return to their piers. Find all possible values of the speed of the rive... | [
"$2\\le v<6$"
] | [
"A motor-boat, whose speed in dead water is 6 km/h, starts from pier $A$ to pier $B$ which is 12 km down the river. At the same time a motor-launch whose speed in dead water is 10 km/h starts from $B$ to $A$. After their meeting, they turn back and return to their piers. Find all possible values of the speed of th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-87c07a234b05a6b0faeb3303340325ebf58c9b6b998201b33d9bc34735245a65 | A motor-cyclist covered the distance of 200 km between towns $A$ and $B$ in 6 h. At first he traveled at the speed $v_1$, exceeding 15 km/h, and then at the speed $v_2$, the lengths of time of movement at each speed being proportional to the speeds. Four hours after he started, the motor-cyclist was 120 km from town $A... | [
"20 km/h and 40 km/h"
] | [
"A motor-cyclist covered the distance of 200 km between towns $A$ and $B$ in 6 h. At first he traveled at the speed $v_1$, exceeding 15 km/h, and then at the speed $v_2$, the lengths of time of movement at each speed being proportional to the speeds. Four hours after he started, the motor-cyclist was 120 km from t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-162c0700b31e8c5a291ae720c4d2299f4fc4232e7257366c64c262b85585b593 | A motor-cyclist covers 1 km 4 min faster than a cyclist. How many km does each of them cover in 5 hours if it is known that during that time the motor-cyclist covers 100 km more than the cyclist? | [
"50 km and 150 km"
] | [
"A motor-cyclist covers 1 km 4 min faster than a cyclist. How many km does each of them cover in 5 hours if it is known that during that time the motor-cyclist covers 100 km more than the cyclist?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7d27a45b36c8435a8ad7437654bd9b25cbf0de39d6fbfa0a576827f7f11bdabf | A motor-cyclist left point $A$ for point $B$, the distance between the points being 120 km. He started back at the same speed, but an hour later he had to stop for 10 min. Then he continued traveling to $A$, having increased the speed by 6 km/h. What was the initial speed of the motor-cyclist if it is known that it too... | [
"48 km/h"
] | [
"A motor-cyclist left point $A$ for point $B$, the distance between the points being 120 km. He started back at the same speed, but an hour later he had to stop for 10 min. Then he continued traveling to $A$, having increased the speed by 6 km/h. What was the initial speed of the motor-cyclist if it is known that ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-18ccfa73139d03fc17e4db011108fbc0da2825d29851e68aae0394f7844945fa | A motor-cyclist left point $A$ for point $B$. Two hours later a car followed him and reached $B$ at the same time as the motor-cyclist. Had the car and the motor-cyclist started from $A$ and $B$ simultaneously to meet each other, then they would have met 1 hour and 20 minutes after they started. How much time does it t... | [
"4 hours"
] | [
"A motor-cyclist left point $A$ for point $B$. Two hours later a car followed him and reached $B$ at the same time as the motor-cyclist. Had the car and the motor-cyclist started from $A$ and $B$ simultaneously to meet each other, then they would have met 1 hour and 20 minutes after they started. How much time does... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cbd78328e040dc190da74cdc1714ff4fb491de9e186b859282c9dbc5afba7a86 | A motor-cyclist starts from point $A$ and travels with a uniform acceleration of 12 km/h$^2$ (the initial speed being zero). Having attained the speed $v$ km/h, be travels with that speed for 25 km and then slows down uniformly, the speed decreasing by 24 km/h, to a complete stop. Then he immediately reverses the motor... | [
"20 km/h"
] | [
"A motor-cyclist starts from point $A$ and travels with a uniform acceleration of 12 km/h$^2$ (the initial speed being zero). Having attained the speed $v$ km/h, be travels with that speed for 25 km and then slows down uniformly, the speed decreasing by 24 km/h, to a complete stop. Then he immediately reverses the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-68972afcd9489bdc00b216e401cd387b922000c66572570a589ef0ca115d9274 | A motorboat is 10 km South of a lighthouse and is on a course of 053 degrees. What is the shortest distance between the motorboat and the lighthouse? | [
"8 km"
] | [
"A motorboat is 10 km South of a lighthouse and is on a course of 053 degrees. What is the shortest distance between the motorboat and the lighthouse?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-cc7a663ea72d0752cb1903d338b4466fc77e35625710c084ed29c3cd30eaa938 | A motorboat is traveling at a constant speed of 10 mi/hr, headed 50 degrees north of west. The river current is 7 mi/hr due west. How far north does the motorboat travel in 3 minutes? | [
"0.383"
] | [
"A motorboat is traveling at a constant speed of 10 mi/hr, headed 50 degrees north of west. The river current is 7 mi/hr due west. How far north does the motorboat travel in 3 minutes?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-705fe4ffe5eefefc06da3d2f199e03dd46b212adb2fb659c2f125c6272f45369 | A motorboat is traveling at a constant speed of 10 mi/hr, headed 50 degrees north of west. The river current is 7 mi/hr due west. How far west does the motorboat travel in 3 minutes? | [
"0.6714"
] | [
"A motorboat is traveling at a constant speed of 10 mi/hr, headed 50 degrees north of west. The river current is 7 mi/hr due west. How far west does the motorboat travel in 3 minutes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-a1f2cae4e16e7eae590084e7bbb198fe2079537a1d3a9edcc71393800ada615f | A motorboat leaves a dock and travels 1370 m due west. Then it turns 24 degrees to the south and travels another 1520 m to a second dock. What is the displacement of the second dock from the first dock? | [
"619"
] | [
"A motorboat leaves a dock and travels 1370 m due west. Then it turns 24 degrees to the south and travels another 1520 m to a second dock. What is the displacement of the second dock from the first dock?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-720e08e6c742bcd7990896a70ea3fca81ea637775128260bd8d3df23eb90a370 | A motorboat traveling with the current went 36 miles in 2 hours. Traveling against the current, it took 3 hours to travel the same distance. Find the rate of the boat in calm water and the rate of the current. | [
"Rate of the boat in calm water = 15 mph, Rate of the current = 3 mph"
] | [
"A motorboat traveling with the current went 36 miles in 2 hours. Traveling against the current, it took 3 hours to travel the same distance. Find the rate of the boat in calm water and the rate of the current."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
omr-ws-3993c3ccc9476757daf4af9abe493b055215e2a607abd3ba1c210a31917a4654 | A motorcycle and a car (Volga) leave $A$ for $B$ at the same time; and at the same time another car (Moskvich) leaves $B$ for $A$. The Moskvich arrives in $A$ in 5 hours and 50 minutes. The cars meet 2 hours 30 minutes later, and the motorcycle and Moskvich meet 140 km from $A$. If the rate of the motorcycle were twice... | [
"Velocities are: motorcycle, 40 km/h, Moskvich car, 60 km/h, and Volga car 80 km/h"
] | [
"A motorcycle and a car (Volga) leave $A$ for $B$ at the same time; and at the same time another car (Moskvich) leaves $B$ for $A$. The Moskvich arrives in $A$ in 5 hours and 50 minutes. The cars meet 2 hours 30 minutes later, and the motorcycle and Moskvich meet 140 km from $A$. If the rate of the motorcycle were ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c64dbbcfb9026fde685fa6639332574a7478e4df4afefa6da8d153412f64fb61 | A motorcyclist and a cyclist simultaneously start out towards each other from points $A$ and $B$ $d$ kilometers apart. In two hours they pass each other and continue on their way. The motorcyclist arrives at $B$ $t$ hours before the cyclist arrives at $A$. Find the speed of the two vehicles. | [
"The speed of the motorcyclist is equal to $d\\frac{t-4+\\sqrt {16+t^2}}{4t}$ km/h, and that of the cyclist, to $d\\frac{t+4-\\sqrt {16+t^2}}{4t}$ km/h"
] | [
"A motorcyclist and a cyclist simultaneously start out towards each other from points $A$ and $B$ $d$ kilometers apart. In two hours they pass each other and continue on their way. The motorcyclist arrives at $B$ $t$ hours before the cyclist arrives at $A$. Find the speed of the two vehicles."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3ceca7548cec6f1e66c5e70128de639fcef1cb5343c6231fc5258e59e11f1700 | A motorist bought 50 litres of petrol every week for 4 consecutive weeks, and the total cost was £163. The price dropped by 2p per litre after the first week. Form an equation involving the price per litre and solve it to find the price per litre in the first week. | [
"0.83"
] | [
"A motorist bought 50 litres of petrol every week for 4 consecutive weeks, and the total cost was £163. The price dropped by 2p per litre after the first week. Form an equation involving the price per litre and solve it to find the price per litre in the first week."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-584f52959eca8ab003a28012b88f30ebb6819c54b445fcfac8306d93b5d880c2 | A motorist travels through 8 sets of traffic lights, each of which is either red or green. He is forced to stop at 3 sets of lights. a) In how many ways could this happen? b) What other number of red lights would give an identical answer to part a)? | [
"a) 56, b) 5"
] | [
"A motorist travels through 8 sets of traffic lights, each of which is either red or green. He is forced to stop at 3 sets of lights.\na) In how many ways could this happen?\nb) What other number of red lights would give an identical answer to part a)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-9ce8ec4ff4ff1d77e51be5b0a4d404512f16332203c82dfc86bb6880b458140a | A motorist travels through 8 sets of traffic lights, each of which is red or green. He is forced to stop at 3 sets of lights. a) In how many ways could this happen? b) What other number of red lights would give an identical answer to part a)? | [
"a) 56, b) 5"
] | [
"A motorist travels through 8 sets of traffic lights, each of which is red or green. He is forced to stop at 3 sets of lights.\na) In how many ways could this happen?\nb) What other number of red lights would give an identical answer to part a)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-c66160259021e1014adf9a8ea345492d87785b6dc258d2e99249229c9e5e11c9 | A mountain climber on one edge of a deep canyon 800 ft wide sees a large rock fall from the opposite edge at time=0. Let \( a \) be the angle of depression of his line of sight below the horizontal. At what angle \( a \) would \( \frac{da}{dt} \) be maximal, given that the rock's acceleration is \(-32 \text{ ft/sec}^2\... | [
"\\(\\frac{\\pi}{6}\\)"
] | [
"A mountain climber on one edge of a deep canyon 800 ft wide sees a large rock fall from the opposite edge at time=0. Let \\( a \\) be the angle of depression of his line of sight below the horizontal. At what angle \\( a \\) would \\( \\frac{da}{dt} \\) be maximal, given that the rock's acceleration is \\(-32 \\te... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-ae1edca6d166664a39846fdc0f6836c8116f428834cb26ea1d1b1ce90d3adf68 | A mountain is of a perfectly conical shape with a base radius of 2 miles and the steepest slopes leading up to the top are 3 miles long. From a point $A$ at the southernmost point of the base, a path leads up on the side of the mountain to $B$, a point on the northern slope and $2/5$ of the way to the top. If $AB$ is t... | [
"\\(\\frac{195}{70}\\)"
] | [
"A mountain is of a perfectly conical shape with a base radius of 2 miles and the steepest slopes leading up to the top are 3 miles long. From a point $A$ at the southernmost point of the base, a path leads up on the side of the mountain to $B$, a point on the northern slope and $2/5$ of the way to the top. If $AB$... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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